Is it possible to trigger share menu on smartphones (via HTML/JS)?

It is possible with a big catch. Currently only available in Chrome for Android, Samsung internet and on Safari (desktop and mobile). And support is coming to Edge and Chrome on desktop http://caniuse.com/#feat=web-share

if (navigator.share) {
  navigator.share({
    title: document.title,
    text: "Hello World",
    url: window.location.href
  })
  .then(() => console.log('Successful share'))
  .catch(error => console.log('Error sharing:', error));
}

https://developers.google.com/web/updates/2016/10/navigator-share